# q-expansion of newform 2100.2.f.h, downloaded from the LMFDB on 19 June 2026.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 2100
weight = 2
poly_data = [1, 0, 3, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0], 1], [[1, 3, 1, 1], 1], [[0, -4, 0, -2], 1], [[2, -3, 1, -1], 1]]

hecke_ring_character_values = [[1051, [1, 0, 0, 0]], [701, [-1, 0, 0, 0]], [1177, [-1, 0, 0, 0]], [1501, [-1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 0], [2, 1, 1, 0], [1, 1, 2, -1], [1, -1, 0, -1], [2, 2, 1, -2], [1, 1, -2, -1], [1, 3, 0, 3], [0, 0, 2, 0], [1, 1, 0, -1], [2, 2, 1, -2], [-8, 2, 0, 2], [1, 1, -2, -1], [-1, -1, 0, 1], [7, 1, 0, 1], [0, -4, 0, -4], [4, 4, 4, -4], [3, 3, 0, -3], [-5, -5, -2, 5], [9, -1, 0, -1], [-6, -2, 0, -2], [1, 1, -2, -1], [8, 2, 0, 2], [-3, -5, 0, -5], [2, -4, 0, -4], [13, -1, 0, -1], [7, -3, 0, -3], [-6, 4, 0, 4], [15, 1, 0, 1], [-9, -9, -4, 9], [2, 6, 0, 6], [-1, 1, 0, 1], [1, 1, -6, -1], [-6, -6, -2, 6], [0, 4, 0, 4], [13, -5, 0, -5], [-7, -7, -2, 7], [1, 1, 6, -1], [0, 0, 1, 0], [-1, -1, -8, 1], [-2, -2, 0, 2], [-1, -1, 2, 1], [0, 0, -3, 0], [23, 1, 0, 1], [-1, -1, -6, 1], [-8, 0, 0, 0], [5, 3, 0, 3], [1, 1, -10, -1], [8, 8, 6, -8], [-13, 5, 0, 5], [5, 5, 8, -5], [-10, -10, -8, 10], [2, -10, 0, -10], [-2, -2, 5, 2], [-11, -5, 0, -5], [-24, 2, 0, 2], [9, 9, 4, -9], [6, 6, 9, -6], [6, 6, 4, -6], [-7, 7, 0, 7], [0, 0, -7, 0], [7, 5, 0, 5], [-10, -10, 0, -10], [-11, 3, 0, 3], [-5, 5, 0, 5], [-12, -4, 0, -4], [0, 0, -1, 0], [-17, -7, 0, -7], [4, 4, 2, -4], [-6, -6, 3, 6], [13, 13, 4, -13], [-9, -7, 0, -7], [2, 2, 7, -2], [-6, 2, 0, 2], [1, 1, -6, -1], [2, 2, 2, -2], [-3, 3, 0, 3], [4, 4, -8, -4], [0, 0, -6, 0], [28, 0, 0, 0], [28, 2, 0, 2], [-1, -1, -2, 1], [17, 7, 0, 7], [-5, -5, -6, 5], [5, -13, 0, -13], [-8, -8, 2, 8], [4, 4, 11, -4], [-20, -2, 0, -2], [-1, -1, -12, 1], [-15, -15, -6, 15], [-4, -4, 0, -4], [-1, -1, 0, 1], [9, 9, -6, -9], [-6, 10, 0, 10], [-7, -7, -2, 7], [14, 4, 0, 4], [10, -8, 0, -8], [-15, 3, 0, 3], [-26, -8, 0, -8], [3, 3, -4, -3], [3, 5, 0, 5], [-13, -13, -12, 13], [-8, -8, -2, 8], [-20, 12, 0, 12], [-3, -13, 0, -13], [3, 3, 8, -3], [-2, -2, -13, 2], [-7, -7, 4, 7], [-14, -14, 0, 14], [11, 13, 0, 13], [-10, -10, -5, 10], [7, -15, 0, -15], [-3, -3, -2, 3], [0, 4, 0, 4], [-6, -6, -16, 6], [-31, 3, 0, 3], [9, 9, 10, -9], [3, -11, 0, -11], [-1, -1, -20, 1], [-12, -12, -8, 12], [-4, -4, 1, 4], [0, 0, 3, 0], [-27, 3, 0, 3], [-5, -5, -4, 5], [4, 4, 8, -4], [14, 4, 0, 4], [24, -8, 0, -8], [3, -3, 0, -3], [33, 7, 0, 7], [-18, -10, 0, -10], [9, 7, 0, 7], [-44, 0, 0, 0], [14, 14, 5, -14], [0, 14, 0, 14], [-6, -6, 6, 6], [0, 0, 9, 0], [23, -3, 0, -3], [-12, -12, -5, 12], [10, 10, 10, -10], [3, 3, 4, -3], [-12, -12, 4, 12], [11, 11, 12, -11], [-33, -3, 0, -3], [-10, -10, -6, 10], [4, 4, 0, 4], [1, -9, 0, -9], [2, 2, -11, -2], [-7, -7, -18, 7], [37, 7, 0, 7], [-2, -2, 13, 2], [10, -12, 0, -12], [11, 11, 0, -11], [-5, -5, 0, 5], [-1, -1, 12, 1], [-13, -13, 2, 13], [30, 2, 0, 2], [2, 8, 0, 8], [17, -1, 0, -1], [-4, -18, 0, -18], [23, 1, 0, 1], [3, -3, 0, -3], [5, 5, -14, -5], [-14, 6, 0, 6], [3, 21, 0, 21], [15, 15, 4, -15], [4, 16, 0, 16], [13, -13, 0, -13], [-4, 6, 0, 6], [-8, -8, 13, 8], [7, 7, 16, -7], [14, 14, 8, -14], [-5, -5, 10, 5], [0, 0, 5, 0], [-5, -5, -6, 5], [-14, -4, 0, -4], [40, -8, 0, -8], [-2, -2, 24, 2], [21, -13, 0, -13], [2, 2, 10, -2], [13, 13, 10, -13], [6, -14, 0, -14], [6, 6, -9, -6], [6, 6, 17, -6], [5, 19, 0, 19], [-18, -4, 0, -4], [-2, -2, 5, 2], [-3, -1, 0, -1], [2, -16, 0, -16], [-46, 2, 0, 2], [-43, -5, 0, -5], [-19, 7, 0, 7], [-32, 8, 0, 8], [26, 8, 0, 8], [3, 5, 0, 5], [14, 14, 23, -14], [-10, -8, 0, -8], [2, 2, 15, -2], [14, 14, 13, -14], [13, 13, -6, -13], [-12, -12, -26, 12], [1, 1, 24, -1], [21, 11, 0, 11], [-2, -2, 22, 2], [-16, -4, 0, -4], [-12, -12, -25, 12], [-5, -5, 6, 5], [1, 3, 0, 3], [6, 6, -18, -6], [-5, -5, -24, 5], [16, 16, -1, -16], [52, 2, 0, 2], [13, 13, -10, -13], [-7, -7, -26, 7], [4, -12, 0, -12], [-2, -2, 32, 2], [3, 3, -16, -3], [8, 14, 0, 14], [3, -3, 0, -3], [-17, -7, 0, -7], [-28, 10, 0, 10], [19, 19, 14, -19], [-2, -2, -18, 2], [5, 5, 30, -5], [-9, -9, 4, 9], [2, 4, 0, 4], [-22, -22, -17, 22], [5, 5, 4, -5], [-1, -11, 0, -11], [9, 9, -2, -9], [-18, -18, -1, 18], [-19, -19, -6, 19], [0, 12, 0, 12], [-6, -6, 8, 6], [5, 23, 0, 23], [3, 3, 24, -3], [-8, -8, 6, 8], [-41, 9, 0, 9], [7, 7, 24, -7], [34, 10, 0, 10], [-7, -13, 0, -13], [-1, -1, -20, 1], [33, -5, 0, -5], [-16, 6, 0, 6], [6, 6, 19, -6], [-48, -8, 0, -8], [15, -7, 0, -7], [-6, -2, 0, -2], [42, 10, 0, 10], [37, -1, 0, -1], [-22, -22, -11, 22], [-40, -2, 0, -2], [-5, 9, 0, 9], [-32, -32, -18, 32], [-12, -12, -11, 12], [-13, -13, 16, 13], [-8, 26, 0, 26], [-29, -11, 0, -11], [0, 0, -17, 0], [-27, 11, 0, 11], [-1, -1, -8, 1], [-37, 13, 0, 13], [8, 8, 22, -8], [-15, 7, 0, 7], [-2, -2, 5, 2], [-7, -7, 22, 7], [20, 20, 10, -20], [-22, -8, 0, -8], [-1, -1, 0, 1], [-13, -11, 0, -11], [-10, -10, -4, 10], [-11, -11, -2, 11], [-55, 7, 0, 7], [-30, 6, 0, 6], [45, 11, 0, 11], [-11, -13, 0, -13], [31, -3, 0, -3], [-18, -20, 0, -20], [-28, 14, 0, 14], [26, -10, 0, -10], [3, 3, -12, -3], [-20, -8, 0, -8], [-13, -13, -20, 13], [-10, -10, 4, 10], [15, 5, 0, 5], [3, 3, 10, -3], [12, 12, 13, -12], [23, 1, 0, 1], [15, 15, 10, -15], [-30, -24, 0, -24], [16, 16, 8, -16], [-11, -11, 18, 11], [-21, -3, 0, -3], [-30, 14, 0, 14], [-14, -14, -17, 14], [-18, 16, 0, 16], [-23, -23, -28, 23], [-8, -8, -27, 8], [-8, -28, 0, -28], [31, -3, 0, -3], [-15, 15, 0, 15], [-29, -3, 0, -3], [-10, 10, 0, 10]]